We finished off the preceding week’s work by solving an inequality involving the absolute value function and a rational function. Also we did an example showing how to bound the absolute value of a rational function on an interval (using the triangle inequality and the reverse triangle inequality).

We began a new chapter: Limits & Continuity. We gave the $\varepsilon$$\delta$ definition of the limit of a function.  We did a few examples. We defined left- and right- hand limits and proved that a limit exists if and only if its left- and right-hand limits exist and are equal.

In one example, we showed a limit did not exist as the left- and right-hand limits did not agree. We then did two examples using the $\varepsilon$$\delta$ definition but noted we wouldn’t always have to do this due to the Calculus of Limits. We wrote down the five results and said we will prove the first and second, put the third and fourth on the webpage, and do a sketch of the fifth.

Problems

You need to do exercises – all of the following you should be able to attempt. Do as many as you can/ want in the following order of most beneficial:

Wills’ Exercise Sheets

More exercise sheets

Q. 3, 4 from Problems

Past Exam Papers

Q . 1(a), 2(a) from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/MS2001s09.pdf

Q. 1(a), 2(a)(i), (b) from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 1(a), 2(a), 3(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 1(a), 2(a), 3 from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 1(a), 2(a), 3 from http://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf